Cyclotron motion in a uniform magnetic field

Equation of motion

In the absence of an electric field, the fundamental equation of motion for a particle of charge q and mass m is given by:

m * dv/dt = q * (v x B)

This force never alters the magnitude of v (it does no work), only its direction.

Parallel/Perpendicular Decomposition

We decompose v = v_para + v_perp, where v_para is the component along B. Since v_para × B = 0, the parallel component remains constant: the motion along B is uniform rectilinear. The perpendicular component, on the other hand, rotates: this is uniform circular motion in the plane perpendicular to B.

Radius and cyclotron frequency

The radius of the circle (Larmor radius) satisfies mv_perp^2/r = qv_perp*B, from which we have:

r = mv_perp/(qB)

The angular frequency, known as the cyclotron frequency, is:

ω_c = q*B/m

Noteworthy point: ω_c does not depend on v_perp, but only on q, B and m. All identical particles rotate at the same frequency, regardless of their velocity: this is cyclotron isochronism, utilised in (cyclotrons).

Overall trajectory

If v_para ≠ 0, the resulting trajectory is a helix with its axis parallel to B, of radius r and pitch p = v_para * (2*pi/omega_c).

Classical trap

Do not confuse the cyclotron frequency ω_c = q*B/m (independent of v) with the angular velocity of any circular motion; bear in mind that only v_perp contributes to the radius r.